By default, the iPhone does not save Drafts, Sent and Deleted(Trash) emails to the server. If you want those emails to be available to all devices that check the server, there are a couple of steps you'll need to do in order to enable that.
- Open "Settings" on your iOS device
- Scroll down the menu and choose "Mail, Contacts, Calendars"
- Select the account you wish to modify
- In the next screen, select your account again
- Choose Advanced at the bottom of the next screen
- Select each folder and choose the corresponding folder on the server to save emails to
- Once you have completed the previous step for each folder, tap "Account" in the upper left of the screen
- Back in your Account screen, tap "Done" in the upper right of the screen
Now your emails will be saved to the server instead of just to your iPhone.